We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), the generalised quantum Singleton bound was believed to hold for many years until recently one of us found a counterexample [MG, arXiv:2007.01249]. Here, we rectify this state of affairs by proving the correct generalised quantum Singleton bound for CQECC, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block length $n$ and code length $k$ for given minimum distance $d$ and we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less than $d$ letters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule, where any pure QECC yields an EAQECC with the same distance and dimension but of shorter block length.

翻译：我们用 von Neumann engropy 不平等的相对简单的推理,可以有力地证明Slenton 量子定型为量子错误校正代码(QECC)。对于缠绕式辅助量子错误校正代码(EAQECC)和催化代码(CQECC),一般量子定型单质代码据信可以维持多年,直到最近才有人发现一个反比[MG, arXiv:2007.01249]。这里,我们通过证明CQECC 的精确通用量定型单质定型,扩展上述QECC的校正法;对于EAQEC 的缠绕式量校正式代码(QEC)和催化代码(C),我们也证明信息-理论性紧凑式紧凑型量子-量子校正码校正码(EAQEC)和催化码交错(C)的交错码(QC QC ) 。所有约束型号都与区段长度($n $: a mess mess made mess made)